On algorithms based on finitely many homomorphism counts

It is a well-known result of Lovász that up to isomorphism a graph G is determined by the homomorphism counts $\text{hom}(F,G)$, i.e., the number of homomorphisms from F to G, where F ranges over all graphs. Thus, in principle, we can answer any query concerning G with only accessing the $\text{hom}(\cdot ,G)$'s instead of G itself. In this paper, we deal with queries φ for which there is a hom algorithm, i.e., there are finitely many graphs $F_1,\dots ,F_k$ such that for any graph G whether it is a Yes-instance of the query is already determined by the vector${\overrightarrow{\text{hom}}}_{F_1,\dots ,F_k}\left(G\right):=\left(\text{hom}\right(F_1,G),\dots ,\text{hom}(F_k,G\left)\right),$ where the graphs $F_1,\dots ,F_k$ only depend on φ.

We observe that planarity of graphs and 3-colorability of graphs, properties expressible in monadic second-order logic, have no hom algorithm. We provide a characterization of the prefix classes of first-order logic with the property that each query definable by a sentence of the prefix class has a hom algorithm.

For adaptive query algorithms, i.e., algorithms that again access ${\overrightarrow{\text{hom}}}_{F_1,\dots ,F_k}\left(G\right)$ but here $F_{i+1}$ might depend on $\text{hom}(F_1,G),\dots ,\text{hom}(F_i,G)$, we show that three homomorphism counts $\text{hom}(\cdot ,G)$ are both sufficient and in general necessary to determine the isomorphism type of G.